- Thread starter
- Moderator
- #1
- Jan 26, 2012
- 995
Thanks again to those who participated in last week's POTW! Here's this week's problem!
-----
Problem: For $x,y>0$, show that
\[\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}= 2\int_0^{\pi/2} \cos^{2x-1}\theta\sin^{2y-1}\theta\,d\theta\]
-----
Hint:
Remember to read the POTW submission guidelines to find out how to submit your answers!
-----
Problem: For $x,y>0$, show that
\[\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}= 2\int_0^{\pi/2} \cos^{2x-1}\theta\sin^{2y-1}\theta\,d\theta\]
-----
Hint:
1) Make the substitution $t=u^2$ in the definition of $\Gamma(x)$, where
\[\Gamma(x) = \int_0^{\infty} e^{-t}t^{x-1}\,dt\]
2) Use (1) to create a double integral for the expression $\Gamma(x)\Gamma(y)$
3) Change to polar coordinates in order to evaluate the double integral.
\[\Gamma(x) = \int_0^{\infty} e^{-t}t^{x-1}\,dt\]
2) Use (1) to create a double integral for the expression $\Gamma(x)\Gamma(y)$
3) Change to polar coordinates in order to evaluate the double integral.
Remember to read the POTW submission guidelines to find out how to submit your answers!